Numerical computation of minimal polynomial bases: A generalized resultant approach

Antoniou, Efstathios N.; Vardulakis, Α. I. G.; Vologiannidis, Stavros

doi:10.1016/j.laa.2005.03.017

Cited by 16 publications

(28 citation statements)

References 16 publications

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“…One well-known example is the study of differential-algebraic equations (see for instance [7] and the references therein). Other sources of problems involving singular matrix polynomials are control and linear systems theory [22], where the problem of computing minimal polynomial bases of null spaces of singular matrix polynomials is still the subject of intense research (see [3] and the references therein for an updated bibliography). In this context, it should be noted that the matrix polynomials arising in control are often full-rank rectangular polynomials.…”

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confidence: 99%

Linearizations of singular matrix polynomials and the recovery of minimal indices

Terán¹,

Dopico²,

Mackey³

2009

ELA

147

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Abstract.A standard way of dealing with a regular matrix polynomial P (λ) is to convert it into an equivalent matrix pencil -a process known as linearization. Two vector spaces of pencils L 1 (P ) and L 2 (P ) that generalize the first and second companion forms have recently been introduced by Mackey, Mackey, Mehl and Mehrmann. Almost all of these pencils are linearizations for P (λ) when P is regular. The goal of this work is to show that most of the pencils in L 1 (P ) and L 2 (P ) are still linearizations when P (λ) is a singular square matrix polynomial, and that these linearizations can be used to obtain the complete eigenstructure of P (λ), comprised not only of the finite and infinite eigenvalues, but also for singular polynomials of the left and right minimal indices and minimal bases. We show explicitly how to recover the minimal indices and bases of the polynomial P (λ) from the minimal indices and bases of linearizations in L 1 (P ) and L 2 (P ). As a consequence of the recovery formulae for minimal indices, we prove that the vector space DL(P ) = L 1 (P ) ∩ L 2 (P ) will never contain any linearization for a square singular polynomial P (λ). Finally, the results are extended to other linearizations of singular polynomials defined in terms of more general polynomial bases.

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confidence: 99%

Linearizations of singular matrix polynomials and the recovery of minimal indices

Terán¹,

Dopico²,

Mackey³

2009

ELA

147

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“…Our algorithm is based on the LQ factorization and has a blocked formulation. We showed that it is more efficient than the similar algorithms presented recently in [23,24]. Moreover, in this paper we presented a full analysis of numerical stability and complexity.…”

Section: Discussionmentioning

confidence: 81%

“…Moreover, the algorithm in [23] does not perform elementary polynomial operations, and the use of standard methods like the SVD improves numerical properties with respect to the classical polynomial methods. A similar work is presented in [24] using matrix resultants. We remark that both papers [23] and [24] appeared after submission of our conference papers [25,26], on which the present paper is based.…”

Section: Brief Review Of Existing Algorithmsmentioning

confidence: 97%

“…We remark that both papers [23] and [24] appeared after submission of our conference papers [25,26], on which the present paper is based. Moreover, as shown in the remainder of this paper, our algorithm performs faster than the algorithms in [23,24] without jeopardizing accuracy and numerical stability.…”

Section: Brief Review Of Existing Algorithmsmentioning

confidence: 97%

“…We also determine simple expressions of the complexity order of our algorithm. A detailed complexity and stability analysis is not carried out for the algorithms proposed in [23] and [24].…”

Section: Contributionmentioning

confidence: 99%

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An improved Toeplitz algorithm for polynomial matrix null-space computation

Anaya

Henrion

2009

Applied Mathematics and Computation

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In this paper we present an improved algorithm to compute the minimal null-space basis of polynomial matrices, a problem which has many applications in control and systems theory. This algorithm takes advantage of the block Toeplitz structure of the Sylvester matrix associated with the polynomial matrix. The analysis of algorithmic complexity and numerical stability shows that the algorithm is reliable and can be considered as an efficient alternative to the well-known pencil (state-space) algorithms found in the literature.

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On Computing Minimal Proper Nullspace Bases with Applications in Fault Detection

Varga¹

2011

Lecture Notes in Electrical Engineering

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We discuss computationally efficient and numerically reliable algorithms to compute minimal proper nullspace bases of a rational or polynomial matrix. The underlying main computational tool is the orthogonal reduction to a Kronecker-like form of the system matrix of an equivalent descriptor system realization. A new algorithm is proposed to compute a simple minimal proper nullspace basis, starting from a non-simple one. Minimal dynamic cover based computational techniques are used for this purpose. The discussed methods allow a high flexibility in addressing several fault detection related applications.

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Numerical computation of minimal polynomial bases: A generalized resultant approach

Cited by 16 publications

References 16 publications

Linearizations of singular matrix polynomials and the recovery of minimal indices

Linearizations of singular matrix polynomials and the recovery of minimal indices

An improved Toeplitz algorithm for polynomial matrix null-space computation

On Computing Minimal Proper Nullspace Bases with Applications in Fault Detection

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